Bayesian Statistics BAYESIAN STATISTICS ∗

Mathematical statistics uses two major paradigms, conventional (or frequentist), and Bayesian. Bayesian methods provide a complete paradigm for both statistical inference and decision making under uncertainty. Bayesian methods may be derived from an axiomatic system, and hence provide a general, coherent methodology. Bayesian methods contain as particular cases many of the more often used frequentist procedures, solve many of the difficulties faced by conventional statistical methods, and extend the applicability of statistical methods. In particular, Bayesian methods make it possible to incorporate scientific hypothesis in the analysis (by means of the prior distribution) and may be applied to problems whose structure is too complex for conventional methods to be able to handle. The Bayesian paradigm is based on an interpretation of probability as a rational, conditional measure of uncertainty, which closely matches the sense of the word ‘probability’ in ordinary language. Statistical inference about a quantity of interest is described as the modification of the uncertainty about its value in the light of evidence, and Bayes’ theorem precisely specifies how this modification should be made. The special situation, often met in scientific reporting and public decision making, where the only acceptable information is that which may be deduced from available documented data, is addressed by objective Bayesian methods, as a particular case.